A Microscopic Model of Black Hole Evaporation in Two Dimensions

We present a microscopic model of black hole (BH) `evaporation' in asymptotically $AdS_2$ spacetimes dual to the low energy sector of the SYK model. To describe evaporation, the SYK model is coupled to a bath comprising of $N_f$ free scalar fields $\Phi_i$. We consider a linear combination of couplings of the form $O_{SYK}(t)\sum_i\Phi_i(0,t)$, where $O_{SYK}$ involves products of the Kourkoulou-Maldacena operator $i J/N\sum_{k=1}^{N/2}s'_k\psi_{2k-1}(t)\psi_{2k}(t)$ specified by a spin vector $s'$. We discuss the time evolution of a product of (i) a pure state of the SYK system, namely a BH microstate characterized by a spin vector $s$ and an effective BH temperature $T_{BH}$, and (ii) a Calabrese-Cardy state of the bath characterized by an effective temperature $T_{bath}$. We take $T_{bath}\ll T_{BH}$, and $T_{BH}$ much lower than the characteristic UV scale $J$ of the SYK model, allowing a description in terms of the time reparameterization mode. Tracing over the bath degrees of freedom leads to a Feynman-Vernon type effective action for the SYK model, which we study in the low energy limit. The leading large $N$ behaviour of the time reparameterization mode is found, as well as the $O(1/\sqrt N)$ fluctuations. The latter are characterized by a non-Markovian non-linear stochastic differential equation with non-local Gaussian noise. In a restricted range of couplings, we find two classes of solutions which asymptotically approach (a) a BH at a lower temperature, and (b) a horizonless geometry. We identify these with partial and complete BH evaporation, respectively. Importantly, the asymptotic solution in both cases involves the scalar product of the spin vectors $s.s'$, which carries some information about the initial state. By repeating the dynamical process $O(N^2)$ times with different choices of the spin vector $s'$, one can in principle reconstruct the initial BH microstate.